I did a lot of computing to test z's up to numbers of 40 digits or more.

I submitted my conjecture in January 1990 to David Allison, who was then the number theory specialist in the Department of Mathematics at the University of Cape Town. He took a great interest in this matter, and wrote to me (16 Jan 1990):"I confess that I am in the same boat as you - I don't see any way to prove this."

It would seem that my conjecture remains a challenge to mathematicians : prove it, or find an example that will disprove it.

I prefer this notation for keyboard use.

It's the activity of the intelligence above all that gives charm to existence.

Re: Gurth's Conjecture

Hi gurthbruins!

Your conjecture seems to be a beautiful piece of hard work!!

I wish that most of the Big Brains here will try untangling this mystery!

I'd request you to tell us all a little about how you happened to stumble upon this nice conjecture.This will give one a Path/Direction to think along; rather than having no clue as to where to start from (ye ye.. i know that most of you actually know where to start from but not everybody else does )!

Re: Gurth's Conjecture

ZHero, I would suggest you start by forgetting about the conjecture and first solve the Diophantine Equation. That would be step one.

If anyone wants to do that, then next he can tackle the job of looking for more solutions. As many as possible. Along the way you will have to learn about Continued Fractions if you do not already know about them. I have seen posts on this forum dealing with Diophantine Equations and Continued Fractions - have a look at those.

I do not know if there are computer programs available these days to crunch large numbers - in 1990 there were none that I could find, I had to write my own to be able to deal with numbers of up to 800 hexadecimal digits. Using machine code for my 6502 microprocessor - computers were slower in those days.

Once you have done sufficient work actually amassing solutions to a problem, you are in a better position to spot patterns in the solutions because you understand more about how it all ticks. I would hardly call this "stumbling", rather making discoveries by thoughtful inspection coupled with understanding of what is going on.

Formulating a conjecture that fits the facts is one thing, proving it quite another. Being quite unable to do that myself, I have no tips to offer. I am not a mathematician, this is just one of my hobbies.

It's the activity of the intelligence above all that gives charm to existence.

Re: Gurth's Conjecture

Hi;

So you had an apple!

For computation sake I manipulated your problem into:

For n = 1,2,3, I could solve getting solutions for x,y, after that it became too difficult. There are many other solutions besides the ones generated by your 2 formulas for n. I am also not sure that it is possible to use continued fractions on this. It is not exactly a pellian. How high were you able to go with n? Can you provide that data?

In mathematics, you don't understand things. You just get used to them.If it ain't broke, fix it until it is.No great discovery was ever made without a bold guess.

Re: Gurth's Conjecture

I had an apple, I enjoyed it, and I threw the core away. Meaning my side of the correspondence with D. Allison.As far as I can remember, I got as high as 8 for n. The corresponding values for x and y went into roughly 200-500 digits, I remember they covered about half a page of foolscap.

I suppose I could enquire at the University if they still have that correspondence, but I am too lazy to do that. But if you want to enquire yourself, you have my permission to do so. You have the date of Allison's last letter, you can refer to that.

I am pretty sure there is no way to do this without Continued Fractions. I was certainly using them all the time.

Of course "There are many other solutions besides the ones generated by your 2 formulas for n." If there hadn't been, it might have been a lot easier to make my conjecture!

It's the activity of the intelligence above all that gives charm to existence.

Explanation of c : I think you must go on with the continued fraction until the cycle of denominators reaches the critical point where it starts to repeat itself. The value of c for z=739 should be 43.

That could be a lot of fun! Want to try it?

It's the activity of the intelligence above all that gives charm to existence.

Re: Gurth's Conjecture

Hi bobbym.

n up to 50 already! Absolutely amazing. Beautiful work!I've copied these 3000 digit values for x and y into a Notepad file - fortunately it is capable of word wrap. (unlike this page). So now you can restore normality to the page.

It's the activity of the intelligence above all that gives charm to existence.